Question

Suppose that a random variableXhas a discrete distribution

with the following p.f.:

\(f\left( x \right) = \left\{ \begin{array}{l}cx\;\;for\;x = 1,...,5,\\0\;\;\;\;otherwise\end{array} \right.\)

Determine the value of the constantc.

Short answer

The value of constant c is \(\frac{1}{{15}}\).

Step-by-step solution

Given information

The random variable X follows the discrete distribution.

The probability function is given as,

\(f\left( x \right) = \left\{ \begin{array}{l}cx;\;\;x = 1,...,5,\\0\;;\;\;\;{\rm{otherwise}}\end{array} \right.\)

Calculate the value for c

It is known that the sum of probabilities over the support of random variable is equal to 1.

From the provided probability function, the value for constant c is computed as,

\(\begin{aligned}{c}\sum\limits_i {f\left( x \right)} & = 1\\\sum\limits_{x = 1}^5 {cx}& = 1\\\left( {1c + 2c + 3c + 4c + 5c} \right)& = 1\\15c &= 1\\c& = \frac{1}{{15}}\end{aligned}\)

Therefore, the value of constant c is \(\frac{1}{{15}}\).